A Diophantine approach to the three and four exponentials conjectures

Abstract

Some special cases of Wilkie’s conjecture are shown to be equivalent to real versions of the three and four exponentials conjectures. Wilkie’s conjecture is an open problem originating in model theory that concerns the density of algebraic points in sets defined using the exponential function; the latter conjectures concern the algebraic nature of values of the exponential function.

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Acknowledgments

Many thanks to the anonymous referee for numerous improvements to the paper including a substantial strengthening of Theorem 9.

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Correspondence to Lee A. Butler.

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The author was supported by the Heilbronn Institute for Mathematical Research.

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Butler, L.A. A Diophantine approach to the three and four exponentials conjectures. Ramanujan J 42, 199–221 (2017). https://doi.org/10.1007/s11139-015-9728-2

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Keywords

  • Transcendence
  • Exponential function
  • Logarithms of algebraic numbers
  • Algebraic independence

Mathematics Subject Classification

  • 03C64
  • 11J81